Evaluate: `lim_(xrarr(pi)/(4)) (sin x -cos x)/(x-(pi)/(4))`

Options

A. `2`
B. `sqrt2`
C. `0`
D. `1`

Explanation

To evaluate the limit

\[ \lim_{x \to \frac{\pi}{4}} \frac{\sin x - \cos x}{x - \frac{\pi}{4}}, \]

we will follow these steps:

### Step 1: Substitute the limit value
First, we substitute \( x = \frac{\pi}{4} \) into the expression to check if we get an indeterminate form:

\[ \sin\left(\frac{\pi}{4}\right) - \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = 0, \]
and
\[ x - \frac{\pi}{4} = \frac{\pi}{4} - \frac{\pi}{4} = 0. \]

Since both the numerator and denominator approach 0, we have an indeterminate form \( \frac{0}{0} \).

### Step 2: Simplify the expression
To resolve the indeterminate form, we can manipulate the expression. We can multiply and divide by \( \frac{1}{\sqrt{2}} \):

\[ \lim_{x \to \frac{\pi}{4}} \frac{\sin x - \cos x}{x - \frac{\pi}{4}} = \lim_{x \to \frac{\pi}{4}} \frac{\frac{1}{\sqrt{2}}(\sin x - \cos x)}{x - \frac{\pi}{4}}. \]

### Step 3: Rewrite the numerator
We can rewrite the numerator using the angle subtraction formula. We know that:

\[ \sin x - \cos x = \sqrt{2} \left( \frac{1}{\sqrt{2}} \sin x - \frac{1}{\sqrt{2}} \cos x \right) = \sqrt{2} \left( \sin x - \sin\left(\frac{\pi}{4}\right) \cos x \right). \]

Thus, we can rewrite our limit as:

\[ \lim_{x \to \frac{\pi}{4}} \frac{\sqrt{2} \left( \sin x - \sin\left(\frac{\pi}{4}\right) \cos x \right)}{x - \frac{\pi}{4}}. \]

### Step 4: Apply L'Hôpital's Rule
Since we still have the form \( \frac{0}{0} \), we can apply L'Hôpital's Rule. We differentiate the numerator and the denominator:

1. The derivative of the numerator \( \sin x - \cos x \) is \( \cos x + \sin x \).
2. The derivative of the denominator \( x - \frac{\pi}{4} \) is \( 1 \).

Thus, we can rewrite the limit as:

\[ \lim_{x \to \frac{\pi}{4}} \frac{\cos x + \sin x}{1}. \]

### Step 5: Substitute again
Now we substitute \( x = \frac{\pi}{4} \):

\[ \cos\left(\frac{\pi}{4}\right) + \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt{2}. \]

### Final Result
Thus, the limit evaluates to:

\[ \lim_{x \to \frac{\pi}{4}} \frac{\sin x - \cos x}{x - \frac{\pi}{4}} = \sqrt{2}. \]